Abstract
Scattering amplitudes in planar super-Yang-Mills theory satisfy several basic physical and mathematical constraints, including physical constraints on their branch cut structure and various empirically discovered connections to the mathematics of cluster algebras. The power of the bootstrap program for amplitudes is inversely proportional to the size of the intersection between these physical and mathematical constraints: ideally we would like a list of constraints which determine scattering amplitudes uniquely. We explore this intersection quantitatively for two-loop six- and seven-point amplitudes by providing a complete taxonomy of the Gr(4, 6) and Gr(4, 7) cluster polylogarithm functions of [15] at weight 4.
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J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic amplitudes and cluster coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].
A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math/0208144] [INSPIRE].
J. Golden and M. Spradlin, A cluster bootstrap for two-loop MHV amplitudes, JHEP 02 (2015) 002 [arXiv:1411.3289] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].
L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr, M. von Hippel and J. Pennington, Bootstrapping six-gluon scattering in planar N = 4 super-Yang-Mills theory, PoS (LL2014) 077 [arXiv:1407.4724] [INSPIRE].
L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].
L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].
S. Caron-Huot and S. He, Jumpstarting the all-loop S-matrix of planar N = 4 super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [INSPIRE].
J.M. Drummond, G. Papathanasiou and M. Spradlin, A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].
A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].
C. Vergu, unpublished.
J. Golden, M.F. Paulos, M. Spradlin and A. Volovich, Cluster polylogarithms for scattering amplitudes, J. Phys. A 47 (2014) 474005 [arXiv:1401.6446] [INSPIRE].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003) 899 [math/0208033].
J.S. Scott, Grassmannians and cluster algebras, Proc. Lond. Math. Soc. 92 (2006) 345.
V.V. Fock and A.B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. 42 (2009) 865 [math/0311245] [INSPIRE].
D. Parker, A. Scherlis, M. Spradlin and A. Volovich, Hedgehog bases for A n cluster polylogarithms and an application to six-point amplitudes, JHEP 11 (2015) 136 [arXiv:1507.01950] [INSPIRE].
F.C.S. Brown, Multiple zeta values and periods of moduli space M 0,n , Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS operators in gauge theories: quivers, syzygies and plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].
A.B. Goncharov, Polylogarithms and motivic Galois group, Proceedings of the Symposium on Pure Mathematics 55, American Mathematical Society, Providence U.S.A. (1994).
S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].
J. Golden and M. Spradlin, An analytic result for the two-loop seven-point MHV amplitude in \( \mathcal{N}=4 \) SYM, JHEP 08 (2014) 154 [arXiv:1406.2055] [INSPIRE].
C. Anastasiou, A. Brandhuber, P. Heslop, V.V. Khoze, B. Spence and G. Travaglini, Two-loop polygon Wilson loops in N = 4 SYM, JHEP 05 (2009) 115 [arXiv:0902.2245] [INSPIRE].
N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, arXiv:1212.5605.
M.F. Paulos and B.U.W. Schwab, Cluster algebras and the positive Grassmannian, JHEP 10 (2014) 31 [arXiv:1406.7273] [INSPIRE].
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Harrington, T., Spradlin, M. Cluster functions and scattering amplitudes for six and seven points. J. High Energ. Phys. 2017, 16 (2017). https://doi.org/10.1007/JHEP07(2017)016
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DOI: https://doi.org/10.1007/JHEP07(2017)016